Mathias:
> Perhaps I might here voice the disquiet felt in certain circles
> over the omission of the name of Richard Taylor in connection with
> FLT.
>> 4. Wiles and Taylor, using an approach previously discarded by Wiles,
> find a new argument which supplants the faulty section of Wiles'
> previous work and finally establishes the truth of FLT.
>Wiles and Taylor chose to present their joint work as two papers --
one a joint paper by Taylor and Wiles (not sure of the order of the
authors) on a key lemma and one with Wiles alone as author proving
TSC (hence FLT) and citing the previous paper. This may be simple
generosity on the part of Taylor, who realized his career was "made"
anyway without requiring a co-author credit, or it may have been
agreed to before Taylor began collaborating with Wiles.
Traditionally, one key issue for deciding the "ownership of the
theorem-credit" is the chronological one -- who proved the last
piece? Obviously Ribet, for example, provided an essential piece but
it wouldn't make sense to give him co-credit for the theorem. My
impression from reading the popular works on Wiles's proof by Singh
and Aczel is that although Taylor made an essential contribution to
the key lemma, it was Wiles who first realized that the work Taylor
had done up to that point was now enough that the proof could be
completed--in other words, he was the first one who "knew" the
theorem was true (in the philosophical sense of justified true
belief--when he though he had finished his earlier proof it was a
case of unjustified true belief but apparently this time all the
steps were correct). If this is the case, then the theorem is
Wiles's alone in this narrow traditional sense. Of course, this line
is difficult to draw in any collaboration -- one person says "I think
I've got it", the other raises an objection, details are thrashed out
together until both are satisfied--who had the last "necessary"
insight?
The other criterion for assigning theorem-credits is
"who did the bulk of the work (since the previous published stage)"?
By this criterion Ribet is out of the picture simply because his
contribution had already been published and credited to him, and
Wiles gets by far the largest share of the credit, but the only
official way of comparing co-authors' contributions is that the first
author did the most etc. and Wiles and Taylor may have felt simply
having him as the first of the two authors may have been unfair to
Wiles.
For all these reasons I am willing to accept the "official"
(published) credit of FLT to Wiles and the key lemma to Wiles and
Taylor (or Taylor and Wiles), even if it turns out that the
popularizations are wrong and it was Taylor who first "knew" the
theorem.
A theorem about which similar issues arose is the proof that all
enumerable sets are Diophantine. Davis, Putnam, and (especially)
Julia Robinson had reduced the question to a simple
number-theoretical hypothesis which Matiyasevich proved. Because
D, P, and R had already published their contributions, M's proof of
the "Julia Robinson hypothesis" in 1970 resulted in his being
assigned the theorem-credit for resolving Hilbert's 10th problem (and
the Fields medal), although some sources give a co-credit to
Chudnovsky who apparently did the same thing independently of
Matiyasevich (perhaps Martin Davis can clear up Chudnovsky's role for
us). In the case of independent discoveries of the same theorem
there is a tradition of giving credit to both discoverers even if one
can be shown to have done it first as long as the other did it before
the first published (typically these independent co-discoveries
occurred within a year of each other, but in today's Internet era
they would have to be a lot closer than that so the phenomenon will
become rarer). Anyway, many now refer to the resolution of Hilbert's
Tenth as the "MDRP" (or some permutation thereof) theorem, which
seems to recognize that although Matiyasevich first "knew" the
theorem the piece he did was small enough relative to the earlier
work that it is more fair to include the others (Y. Manin in his book
"A Course in Mathematical Logic" credits in order Davis, Putnam,
Robinson, Matiyasevich, and Chudnovsky; this order appears to be
chronological but since M's contribution was sufficient it suggests
C's was a slightly later independent codiscovery).
Another case is the discovery of NP-completeness, originally
attributed to Cook and Karp but now revised to co-credit Leonid
Levin. (Can Steve Cook confirm that his discovery and Karp's were
independent?) It is to be hoped that mathematical communication with
and within Russia is so much better now that we won't have such confusion
for future important theorems (the old Tomsk-to-Omsk-to-Minsk-to-Pinsk
route popularized by Tom Lehrer in his song "Lobachevsky" reflected
an unfortunate reality in the 60's!).
Joe Shipman